The house was purchased with Cash using HELOC funds from our primary residence.

Sure, if you have a HELOC with a checkbook to access the funds, you can write a check for the EMD. 

This is my first rental property purchase, and I was deceived into thinking I had to wire transfer funds for closing in order to make my closing date on the 20th.

There is the week-ish delay of getting your funds but you should have reserves for repairs etc. anyway so that shouldn't be a problem. 

It's just me, my wife and my kids.We are considering looking for and moving to a single family home in a nearby area and I was wondering if we could go through the same or similar process where my parents would 1031 exchange their ownership in the current property towards a new property and where we would sell our ownership and apply the funds towards the downpayment, etc on the new property accordingly.

Obviously that was funded 100% lender had nothing to lose . those two deals alone are equal to what most make in a dozen or so deals.

I have come to believe I should have a substantial portion of my net worth invested in a Schwab Total Stock Market Index Fund (SWTSX) or Vanguard Total Stock Market Index Fund (VTSAX).

I'm currently in the very early stages of trying to roll an old 401k into an SDIRA and use the $22k worth of funds to get me involved in my first deal.

Part of these funds has already been ear marked for renovating our house to create an Airbnb unit, which will help the financial freedom goal, but that's not the focus of this question...

Used some of the HELOC funds to buy a SF that needed some work in which I moved into for a live & flip.